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Completeness and semi-flows for stochastic differential equations with monotone drift

Michael Scheutzow Technische Universit¨ at Berlin Bielefeld, October 4th, 2012

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Differential Equations

An SDE dXt = b(Xt) dt + Pm

j=1 j(Xt) dWj(t), X0 = x 2 Rd.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Differential Equations

An SDE dXt = b(Xt) dt + Pm

j=1 j(Xt) dWj(t), X0 = x 2 Rd.

Well-known Existence and uniqueness of solutions, continuous dependence on initial condition and existence of solution flow of homeomorphisms if b, globally Lipschitz.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Differential Equations

An SDE dXt = b(Xt) dt + Pm

j=1 j(Xt) dWj(t), X0 = x 2 Rd.

Well-known Existence and uniqueness of solutions, continuous dependence on initial condition and existence of solution flow of homeomorphisms if b, globally Lipschitz. Are these properties still true (at least locally) in case infinite number of driving BM (or Kunita-type sdes) local one-sided Lipschitz condition?

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Existence and uniqueness of local solutions

Consider the sde dXt = b(Xt) dt + M(dt, Xt), X0 = x 2 Rd, where b continuous, M cont. martingale field s.t. a(x, y) := d

dt [M(t, x), M(t, y)] is cont. and determ.,

A(x, y) :=a(x, x)

• a(x, y)
• a(y, x)

+ a(y, y)(= d

dt [M(t, x)M(t, y)])

and One-sided local Lipschitz condition 2hb(x) b(y), x yi + TrA(x, y)  KN|x y|2, |x|, |y|  N.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Existence and uniqueness of local solutions

Consider the sde dXt = b(Xt) dt + M(dt, Xt), X0 = x 2 Rd, where b continuous, M cont. martingale field s.t. a(x, y) := d

dt [M(t, x), M(t, y)] is cont. and determ.,

A(x, y) :=a(x, x)

• a(x, y)
• a(y, x)

+ a(y, y)(= d

dt [M(t, x)M(t, y)])

and One-sided local Lipschitz condition 2hb(x) b(y), x yi + TrA(x, y)  KN|x y|2, |x|, |y|  N. Theorem The sde has a unique local solution.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Existence and uniqueness of local solutions

Consider the sde dXt = b(Xt) dt + M(dt, Xt), X0 = x 2 Rd, where b continuous, M cont. martingale field s.t. a(x, y) := d

dt [M(t, x), M(t, y)] is cont. and determ.,

A(x, y) :=a(x, x)

• a(x, y)
• a(y, x)

+ a(y, y)(= d

dt [M(t, x)M(t, y)])

and One-sided local Lipschitz condition 2hb(x) b(y), x yi + TrA(x, y)  KN|x y|2, |x|, |y|  N. Theorem The sde has a unique local solution. In the “classical” case M(t, x) = Pm

j=1 j(x)Wj(t)

A(x, y) = ((x)(y))((x)(y))t, TrA(x, y) = k(x)(y)k2

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Idea of proof (Krylov, Pr´ evˆ

• t/R¨
• ckner): Euler approx.

For n 2 N let n

0 := x and for t 2 ( k n, k+1 n ]:

n

t := n k/n +

R t

k/n b(n k/n) ds +

R t

k/n M(ds, n k/n).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Idea of proof (Krylov, Pr´ evˆ

• t/R¨
• ckner): Euler approx.

For n 2 N let n

0 := x and for t 2 ( k n, k+1 n ]:

n

t := n k/n +

R t

k/n b(n k/n) ds +

R t

k/n M(ds, n k/n).

So, Up to appropriate stopping time: |n

t m t |2  ...  2KR

R t

0 |n s m s |2 ds +

R t

0 sth. small ds + Nt

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Idea of proof (Krylov, Pr´ evˆ

• t/R¨
• ckner): Euler approx.

For n 2 N let n

0 := x and for t 2 ( k n, k+1 n ]:

n

t := n k/n +

R t

k/n b(n k/n) ds +

R t

k/n M(ds, n k/n).

So, Up to appropriate stopping time: |n

t m t |2  ...  2KR

R t

0 |n s m s |2 ds +

R t

0 sth. small ds + Nt

Now use Stochastic Gronwall Lemma (v. Renesse, S., 2010) Let Z 0, H be adapted cts., N cts. local mart, N0 = 0 s.t. Zt  K R t

0 Z ⇤ u du + Nt + Ht, t 0.

Then, for each 0 < p < 1 and ↵ > 1+p

1p 9c1, c2:

E(Z ⇤

T)p  c1 exp{c2KT}(EH⇤α T )p/α.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Existence of global solutions

Theorem If, in addition, there exists a nondecr. ⇢ : [0, 1) ! (0, 1) s.t. R 1

0 1/⇢(u) du = 1 and

2hb(x), xi + Tr(a(x, x))  ⇢(|x|2), x 2 Rd, then the local solution of the sde is global ((weakly) complete).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Existence of global solutions

Theorem If, in addition, there exists a nondecr. ⇢ : [0, 1) ! (0, 1) s.t. R 1

0 1/⇢(u) du = 1 and

2hb(x), xi + Tr(a(x, x))  ⇢(|x|2), x 2 Rd, then the local solution of the sde is global ((weakly) complete). Itˆ

• ’s formula implies

X 2

τ X 2 0 =

Z τ 2hb(Xu), Xui + Tr(a(Xu, Xu)) du + Nτ  Z τ ⇢(|Xu|2) du + Nτ.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Existence of global solutions

Theorem If, in addition, there exists a nondecr. ⇢ : [0, 1) ! (0, 1) s.t. R 1

0 1/⇢(u) du = 1 and

2hb(x), xi + Tr(a(x, x))  ⇢(|x|2), x 2 Rd, then the local solution of the sde is global ((weakly) complete). Itˆ

• ’s formula implies

X 2

τ X 2 0 =

Z τ 2hb(Xu), Xui + Tr(a(Xu, Xu)) du + Nτ  Z τ ⇢(|Xu|2) du + Nτ. Lemma (v. Renesse, S., 2010) Let Z 0 be adapted cts. defined on [0, ), N cts. local mart, N0 = 0, C 0 s.t. Zt  R t

0 ⇢(Z ⇤ u ) du + Nt + C, t 2 [0, )

and limt"σ Z ⇤

t = 1 on { < 1}. Then = 1 almost surely.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous dependence on initial condition

Question Are our conditions sufficient for Continuous dependence on initial conditions (or even the semi-flow property)? In particular: Do conditions for global existence of solutions ensure existence of a continuous map ' : [0, 1) ⇥ Rd ! Rd which is a modification of the solution map (strong completeness)?

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous dependence on initial condition

Question Are our conditions sufficient for Continuous dependence on initial conditions (or even the semi-flow property)? In particular: Do conditions for global existence of solutions ensure existence of a continuous map ' : [0, 1) ⇥ Rd ! Rd which is a modification of the solution map (strong completeness)? Answer: No!

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous dependence on initial condition

Question Are our conditions sufficient for Continuous dependence on initial conditions (or even the semi-flow property)? In particular: Do conditions for global existence of solutions ensure existence of a continuous map ' : [0, 1) ⇥ Rd ! Rd which is a modification of the solution map (strong completeness)? Answer: No! There exists a 2d sde without drift driven by a single BM with bounded and C1 coefficient which is not strongly complete. Reference: Li, S.: Lack of strong completeness .. (Ann. Prob. 2011)

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous modification

Lemma Assume that for some µ, K 0, and all x, y 2 Rd 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  K|x y|2. Then the sde is weakly complete. Denote solutions by t(x). For each q 2 (0, µ + 2), there exist c1, c2 s.t. E sup0sT |s(x) s(y)|q  c1|x y|q exp{c2KT} holds for all x, y, T.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous modification

Lemma Assume that for some µ, K 0, and all x, y 2 Rd 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  K|x y|2. Then the sde is weakly complete. Denote solutions by t(x). For each q 2 (0, µ + 2), there exist c1, c2 s.t. E sup0sT |s(x) s(y)|q  c1|x y|q exp{c2KT} holds for all x, y, T. Proof Show: above condition implies suff. cond. for global existence. Define Zt := |t(x) t(y)|µ+2. Then dZt = ... and Zt  |x y|µ+2 + ( µ

2 + 1)K

R t

0 Zs ds + Nt.

Applying Stochastic Gronwall Lemma yields assertion.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous modification

Lemma Assume that for some µ 0, and nondecr. f : [0, 1) ! (0, 1) 2hb(x)b(y), xyi+TrA(x, y)+µkA(x, y)k  f(|x| _ |y|)|xy|2. Assume the sde is weakly complete. Then for q 2 (0, µ + 2), " > 0 and B := (µ+2)(1+ε)

µ+2q

: E sup

0sT

|s(x) s(y)|q  cq,ε,µ|xy|q(E exp{qB 2 Z T f(|s(x)| _ |s(y)|) ds})1/B  cq,ε,µ|xy|q max

z2{x,y}(E exp{qB

Z T f(|s(z)|) ds})1/B  ...

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Continuous modification

Lemma Assume that for some µ 0, and nondecr. f : [0, 1) ! (0, 1) 2hb(x)b(y), xyi+TrA(x, y)+µkA(x, y)k  f(|x| _ |y|)|xy|2. Assume the sde is weakly complete. Then for q 2 (0, µ + 2), " > 0 and B := (µ+2)(1+ε)

µ+2q

: E sup

0sT

|s(x) s(y)|q  cq,ε,µ|xy|q(E exp{qB 2 Z T f(|s(x)| _ |s(y)|) ds})1/B  cq,ε,µ|xy|q max

z2{x,y}(E exp{qB

Z T f(|s(z)|) ds})1/B  ... Corollary If the ass. holds for some µ > d 2 and the last expectation is locally bounded in z for some q > d and T > 0, then the sde is strongly complete.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Criteria for strong completeness

2hb(x), xi + Tr(a(x, x))  c(1 + |x|)2, f(x) = log+ x 2hb(x), xi, Tr(a(x, x))  c(1 + |x|)2, f(x) = (log+ x)2 b, a bounded f(x) = x2

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Criteria for strong completeness

2hb(x), xi + Tr(a(x, x))  c(1 + |x|)2, f(x) = log+ x 2hb(x), xi, Tr(a(x, x))  c(1 + |x|)2, f(x) = (log+ x)2 b, a bounded f(x) = x2 NB If, for some µ > d 2, we have 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  KN|x y|2, then the sde has a locally continuous modification (up to explosion).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Semiflow

Lemma Assume that a, b are bounded and (?) 2hb(x)b(y), xyi+TrA(x, y)+µkA(x, y)k  K|xy|2 for some µ 0. Then the sde is complete and for q 2 (0, µ + 2) and T > 0 there is c > 0 s.t. E|st(x) s0t0(y)|q  c(|x y|q + |s s0|q/2 + |t t0|q/2) for all 0  s, s0, t, t0  T.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Semiflow

Lemma Assume that a, b are bounded and (?) 2hb(x)b(y), xyi+TrA(x, y)+µkA(x, y)k  K|xy|2 for some µ 0. Then the sde is complete and for q 2 (0, µ + 2) and T > 0 there is c > 0 s.t. E|st(x) s0t0(y)|q  c(|x y|q + |s s0|q/2 + |t t0|q/2) for all 0  s, s0, t, t0  T. Proposition If (?) holds for some µ > d + 2, then the sde generates a global semi-flow.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Semiflow

Lemma Assume that a, b are bounded and (?) 2hb(x)b(y), xyi+TrA(x, y)+µkA(x, y)k  K|xy|2 for some µ 0. Then the sde is complete and for q 2 (0, µ + 2) and T > 0 there is c > 0 s.t. E|st(x) s0t0(y)|q  c(|x y|q + |s s0|q/2 + |t t0|q/2) for all 0  s, s0, t, t0  T. Proposition If (?) holds for some µ > d + 2, then the sde generates a global semi-flow. If (?) holds for some µ > d + 2 locally, then the sde generates a local semi-flow.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Stochastic Semiflow

Lemma Assume that a, b are bounded and (?) 2hb(x)b(y), xyi+TrA(x, y)+µkA(x, y)k  K|xy|2 for some µ 0. Then the sde is complete and for q 2 (0, µ + 2) and T > 0 there is c > 0 s.t. E|st(x) s0t0(y)|q  c(|x y|q + |s s0|q/2 + |t t0|q/2) for all 0  s, s0, t, t0  T. Proposition If (?) holds for some µ > d + 2, then the sde generates a global semi-flow. If (?) holds for some µ > d + 2 locally, then the sde generates a local semi-flow. Local semi-flow + strong completeness ) global semi-flow.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong p-completeness

Possible Definitions Let p 2 [0, d] and assume either nothing sde has locally continuous modif. ' sde generates local semi-flow '. Then the sde is called strongly p-complete if for every A ⇢ Rd

• f dimension at most p there exists a modif. ' of the local

solution which restricted to A is continuous (Rd-valued) in (t, x), where dimension may stand for either Hausdorff dimension Upper Minkowski dimension (= box dimension) something else

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong p-completeness

Possible Definitions Let p 2 [0, d] and assume either nothing sde has locally continuous modif. ' sde generates local semi-flow '. Then the sde is called strongly p-complete if for every A ⇢ Rd

• f dimension at most p there exists a modif. ' of the local

solution which restricted to A is continuous (Rd-valued) in (t, x), where dimension may stand for either Hausdorff dimension Upper Minkowski dimension (= box dimension) something else NB In local semi-flow case d 1-completeness implies d-completeness.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Proposition Let p 2 [0, d] and assume that for some µ > p 2 we have 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  K|x y|2. Then, the sde is strongly p-complete in the “nothing/upper Minkowski” sense.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Proposition Let p 2 [0, d] and assume that for some µ > p 2 we have 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  K|x y|2. Then, the sde is strongly p-complete in the “nothing/upper Minkowski” sense. Proof For q 2 (p, µ + 2) we saw that E sup0sT |s(x) s(y)|q  c1|x y|q exp{c2KT}. Theorem 11.1/11.6 in Ledoux-Talagrand applied to a set A ⇢ Rd of upper Mink. dim. p implies the claim.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Proposition Let p 2 [0, d] and assume that for some µ > p 2 we have 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  K|x y|2. Then, the sde is strongly p-complete in the “nothing/upper Minkowski” sense. Proof For q 2 (p, µ + 2) we saw that E sup0sT |s(x) s(y)|q  c1|x y|q exp{c2KT}. Theorem 11.1/11.6 in Ledoux-Talagrand applied to a set A ⇢ Rd of upper Mink. dim. p implies the claim. NB In previous prop. the image of A is even bounded for each t > 0.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Proposition Let p 2 [0, d] and assume that for some µ > p 2 we have 2hb(x) b(y), x yi + TrA(x, y) + µkA(x, y)k  K|x y|2. Then, the sde is strongly p-complete in the “nothing/upper Minkowski” sense. Proof For q 2 (p, µ + 2) we saw that E sup0sT |s(x) s(y)|q  c1|x y|q exp{c2KT}. Theorem 11.1/11.6 in Ledoux-Talagrand applied to a set A ⇢ Rd of upper Mink. dim. p implies the claim. NB In previous prop. the image of A is even bounded for each t > 0. Question Is Proposition true with “local semi-flow/Hausdorff”?

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: a different approach

dX(t) = b(X(t)) dt + M(dt, X(t)), with b locally Lipschitz and M 2 B1,δ

ub .

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: a different approach

dX(t) = b(X(t)) dt + M(dt, X(t)), with b locally Lipschitz and M 2 B1,δ

ub . Let be the (strongly

complete) flow ass. to dY(t) = M(dt, Y(t)).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: a different approach

dX(t) = b(X(t)) dt + M(dt, X(t)), with b locally Lipschitz and M 2 B1,δ

ub . Let be the (strongly

complete) flow ass. to dY(t) = M(dt, Y(t)). F(t, z, x, !) := {D (t, z, !)}1b(x) ⇣(t, x, !) := (t, ., !)1(x), t 0, x, z 2 Rd Then X(t, !) = (t, x + R t

0 F(u, ⇣(u, X(u, !), !), X(u, !), !) du, !).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: a different approach

dX(t) = b(X(t)) dt + M(dt, X(t)), with b locally Lipschitz and M 2 B1,δ

ub . Let be the (strongly

complete) flow ass. to dY(t) = M(dt, Y(t)). F(t, z, x, !) := {D (t, z, !)}1b(x) ⇣(t, x, !) := (t, ., !)1(x), t 0, x, z 2 Rd Then X(t, !) = (t, x + R t

0 F(u, ⇣(u, X(u, !), !), X(u, !), !) du, !).

If either there exists 2 (0, 1) s.t. |b(x)|  C(1 + |x|γ) or |b(x)|  C(1 + |x|), supx,u kD (u, x, !)1k < 1, then the SDE is strongly complete (Mohammed, S., JFA, ’03).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: Conjecture and Counterexample

Conjecture b linear growth, noise globally Lip. ) strong completeness.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: Conjecture and Counterexample

Conjecture b linear growth, noise globally Lip. ) strong completeness. Stronger Conjecture b linear growth, noise globally Lip. ) images of bounded sets grow at most exponentially (with deterministic rate).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: Conjecture and Counterexample

Conjecture b linear growth, noise globally Lip. ) strong completeness. Stronger Conjecture b linear growth, noise globally Lip. ) images of bounded sets grow at most exponentially (with deterministic rate). But: b linear growth in radial dir., noise globally Lipschitz ; strong completeness.

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Strong Completeness: Conjecture and Counterexample

Conjecture b linear growth, noise globally Lip. ) strong completeness. Stronger Conjecture b linear growth, noise globally Lip. ) images of bounded sets grow at most exponentially (with deterministic rate). But: b linear growth in radial dir., noise globally Lipschitz ; strong completeness. Even: hb(x), xi = 0, noise globally Lipschitz and bounded ; strong completeness. (Example on blackboard)

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations

Random Dynamical Systems

Proposition If the SDE generates a global semi-flow, then it also generates a random dynamical system. Proof Follows as in Kager-S. (EJP , 1997).

Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations